\(\int \frac {(-1+x^3)^{2/3}}{x^4} \, dx\) [1262]

Optimal result

Integrand size = 13, antiderivative size = 92 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^4} \, dx=-\frac {\left (-1+x^3\right )^{2/3}}{3 x^3}-\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2}{9} \log \left (1+\sqrt [3]{-1+x^3}\right )+\frac {1}{9} \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \] Output:

-1/3*(x^3-1)^(2/3)/x^3+2/9*arctan(-1/3*3^(1/2)+2/3*(x^3-1)^(1/3)*3^(1/2))* 
3^(1/2)-2/9*ln(1+(x^3-1)^(1/3))+1/9*ln(1-(x^3-1)^(1/3)+(x^3-1)^(2/3))
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.90 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^4} \, dx=\frac {1}{9} \left (-\frac {3 \left (-1+x^3\right )^{2/3}}{x^3}-2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )-2 \log \left (1+\sqrt [3]{-1+x^3}\right )+\log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \] Input:

Integrate[(-1 + x^3)^(2/3)/x^4,x]
 

Output:

((-3*(-1 + x^3)^(2/3))/x^3 - 2*Sqrt[3]*ArcTan[(1 - 2*(-1 + x^3)^(1/3))/Sqr 
t[3]] - 2*Log[1 + (-1 + x^3)^(1/3)] + Log[1 - (-1 + x^3)^(1/3) + (-1 + x^3 
)^(2/3)])/9
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.80, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {798, 51, 68, 16, 1083, 217}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (x^3-1\right )^{2/3}}{x^4} \, dx\)

\(\Big \downarrow \) 798

\(\displaystyle \frac {1}{3} \int \frac {\left (x^3-1\right )^{2/3}}{x^6}dx^3\)

\(\Big \downarrow \) 51

\(\displaystyle \frac {1}{3} \left (\frac {2}{3} \int \frac {1}{x^3 \sqrt [3]{x^3-1}}dx^3-\frac {\left (x^3-1\right )^{2/3}}{x^3}\right )\)

\(\Big \downarrow \) 68

\(\displaystyle \frac {1}{3} \left (\frac {2}{3} \left (-\frac {3}{2} \int \frac {1}{\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}+\frac {3}{2} \int \frac {1}{x^6-\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}+\frac {\log \left (x^3\right )}{2}\right )-\frac {\left (x^3-1\right )^{2/3}}{x^3}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{3} \left (\frac {2}{3} \left (\frac {3}{2} \int \frac {1}{x^6-\sqrt [3]{x^3-1}+1}d\sqrt [3]{x^3-1}+\frac {\log \left (x^3\right )}{2}-\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )-\frac {\left (x^3-1\right )^{2/3}}{x^3}\right )\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {1}{3} \left (\frac {2}{3} \left (-3 \int \frac {1}{-x^6-3}d\left (2 \sqrt [3]{x^3-1}-1\right )+\frac {\log \left (x^3\right )}{2}-\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )-\frac {\left (x^3-1\right )^{2/3}}{x^3}\right )\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {1}{3} \left (\frac {2}{3} \left (\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^3-1}-1}{\sqrt {3}}\right )+\frac {\log \left (x^3\right )}{2}-\frac {3}{2} \log \left (\sqrt [3]{x^3-1}+1\right )\right )-\frac {\left (x^3-1\right )^{2/3}}{x^3}\right )\)

Input:

Int[(-1 + x^3)^(2/3)/x^4,x]
 

Output:

(-((-1 + x^3)^(2/3)/x^3) + (2*(Sqrt[3]*ArcTan[(-1 + 2*(-1 + x^3)^(1/3))/Sq 
rt[3]] + Log[x^3]/2 - (3*Log[1 + (-1 + x^3)^(1/3)])/2))/3)/3
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 68

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ 
{q = Rt[-(b*c - a*d)/b, 3]}, Simp[Log[RemoveContent[a + b*x, x]]/(2*b*q), x 
] + (Simp[3/(2*b)   Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/3)], 
 x] - Simp[3/(2*b*q)   Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] / 
; FreeQ[{a, b, c, d}, x] && NegQ[(b*c - a*d)/b]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 798

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n   Subst 
[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, 
b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.04

method	result	size
risch	\(-\frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{3 x^{3}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+3 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{9 \pi \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\)	\(96\)
meijerg	\(\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-\frac {\pi \sqrt {3}\, x^{3} \operatorname {hypergeom}\left (\left [1, 1, \frac {4}{3}\right ], \left [2, 3\right ], x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}-1+3 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right ) x^{3}}\right )}{9 \pi {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}}}\)	\(97\)
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{3}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right ) x^{3}+\ln \left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right ) x^{3}-2 \ln \left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right ) x^{3}-3 \left (x^{3}-1\right )^{\frac {2}{3}}}{9 \left (1-\left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right ) \left (1+\left (x^{3}-1\right )^{\frac {1}{3}}\right )}\)	\(107\)
trager	\(-\frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{3 x^{3}}+\frac {16 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \ln \left (-\frac {92032 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{3}-78840 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{3}+44016 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-1477 x^{3}+14247 \left (x^{3}-1\right )^{\frac {2}{3}}-44016 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}-736256 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}-14247 \left (x^{3}-1\right )^{\frac {1}{3}}+136048 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )+2743}{x^{3}}\right )}{9}+\frac {2 \ln \left (\frac {-92032 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{3}-55832 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{3}+44016 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+9894 x^{3}-19749 \left (x^{3}-1\right )^{\frac {2}{3}}-44016 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+736256 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}+19749 \left (x^{3}-1\right )^{\frac {1}{3}}-48016 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-8245}{x^{3}}\right )}{9}-\frac {16 \ln \left (\frac {-92032 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2} x^{3}-55832 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) x^{3}+44016 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+9894 x^{3}-19749 \left (x^{3}-1\right )^{\frac {2}{3}}-44016 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}+736256 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )^{2}+19749 \left (x^{3}-1\right )^{\frac {1}{3}}-48016 \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )-8245}{x^{3}}\right ) \operatorname {RootOf}\left (64 \textit {\_Z}^{2}-8 \textit {\_Z} +1\right )}{9}\)	\(439\)

Input:

int((x^3-1)^(2/3)/x^4,x,method=_RETURNVERBOSE)
 

Output:

-1/3*(x^3-1)^(2/3)/x^3+1/9/Pi*3^(1/2)*GAMMA(2/3)/signum(x^3-1)^(1/3)*(-sig 
num(x^3-1))^(1/3)*(2/9*Pi*3^(1/2)/GAMMA(2/3)*x^3*hypergeom([1,1,4/3],[2,2] 
,x^3)+2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x)+I*Pi)*Pi*3^(1/2)/GAMMA(2/3))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.87 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^4} \, dx=\frac {2 \, \sqrt {3} x^{3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + x^{3} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) - 2 \, x^{3} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) - 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{9 \, x^{3}} \] Input:

integrate((x^3-1)^(2/3)/x^4,x, algorithm="fricas")
 

Output:

1/9*(2*sqrt(3)*x^3*arctan(2/3*sqrt(3)*(x^3 - 1)^(1/3) - 1/3*sqrt(3)) + x^3 
*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) - 2*x^3*log((x^3 - 1)^(1/3) + 
1) - 3*(x^3 - 1)^(2/3))/x^3
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.63 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.35 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^4} \, dx=- \frac {\Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x \Gamma \left (\frac {4}{3}\right )} \] Input:

integrate((x**3-1)**(2/3)/x**4,x)
 

Output:

-gamma(1/3)*hyper((-2/3, 1/3), (4/3,), exp_polar(2*I*pi)/x**3)/(3*x*gamma( 
4/3))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^4} \, dx=\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, x^{3}} + \frac {1}{9} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{9} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) \] Input:

integrate((x^3-1)^(2/3)/x^4,x, algorithm="maxima")
 

Output:

2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) - 1/3*(x^3 - 1)^(2 
/3)/x^3 + 1/9*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) - 2/9*log((x^3 - 
1)^(1/3) + 1)
 

Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^4} \, dx=\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{3 \, x^{3}} + \frac {1}{9} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {2}{9} \, \log \left ({\left | {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \] Input:

integrate((x^3-1)^(2/3)/x^4,x, algorithm="giac")
 

Output:

2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^3 - 1)^(1/3) - 1)) - 1/3*(x^3 - 1)^(2 
/3)/x^3 + 1/9*log((x^3 - 1)^(2/3) - (x^3 - 1)^(1/3) + 1) - 2/9*log(abs((x^ 
3 - 1)^(1/3) + 1))
 

Mupad [B] (verification not implemented)

Time = 9.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^4} \, dx=-\frac {2\,\ln \left (\frac {4\,{\left (x^3-1\right )}^{1/3}}{9}+\frac {4}{9}\right )}{9}-\ln \left (9\,{\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2+\frac {4\,{\left (x^3-1\right )}^{1/3}}{9}\right )\,\left (-\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )+\ln \left (9\,{\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )}^2+\frac {4\,{\left (x^3-1\right )}^{1/3}}{9}\right )\,\left (\frac {1}{9}+\frac {\sqrt {3}\,1{}\mathrm {i}}{9}\right )-\frac {{\left (x^3-1\right )}^{2/3}}{3\,x^3} \] Input:

int((x^3 - 1)^(2/3)/x^4,x)
                                                                                    
                                                                                    
 

Output:

log(9*((3^(1/2)*1i)/9 + 1/9)^2 + (4*(x^3 - 1)^(1/3))/9)*((3^(1/2)*1i)/9 + 
1/9) - log(9*((3^(1/2)*1i)/9 - 1/9)^2 + (4*(x^3 - 1)^(1/3))/9)*((3^(1/2)*1 
i)/9 - 1/9) - (2*log((4*(x^3 - 1)^(1/3))/9 + 4/9))/9 - (x^3 - 1)^(2/3)/(3* 
x^3)
 

Reduce [F]

\[ \int \frac {\left (-1+x^3\right )^{2/3}}{x^4} \, dx=\frac {-\left (x^{3}-1\right )^{\frac {2}{3}}+2 \left (\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{4}-x}d x \right ) x^{3}}{3 x^{3}} \] Input:

int((x^3-1)^(2/3)/x^4,x)
 

Output:

( - (x**3 - 1)**(2/3) + 2*int((x**3 - 1)**(2/3)/(x**4 - x),x)*x**3)/(3*x** 
3)